Spectral Gap for the Binary Fixed-Margin Swap Chain

We prove an inverse-polynomial spectral-gap bound for the lazy swap chain on binary matrices with prescribed row and column sums. This chain is a standard sampler for fixed-margin null models in ecology, statistics, and network analysis, and its rapid mixing for arbitrary feasible margins was conjectured by Kannan, Tetali, and Vempala in 1997. We show that for every feasible set of margins on an $m\times n$ binary matrix, the lazy swap chain has spectral gap at least $$ \binom{m}{2}^{-1}\binom{n}{2}^{-1}, $$ which is tight in the worst case. The proof compares the swap chain with a two-row heat-bath chain, reduces the analysis from arbitrary $m\times n$ matrices to the case of three rows, and proves the resulting three-row inequality by decomposing functions according to the column-count variable and the associated Johnson harmonic sectors. The proof itself was generated by ChatGPT 5.5 Pro. ChatGPT proposed the whole proof strategy, including the comparison with the two-row heat-bath chain, the reduction to the three-row case, and the decomposition of the three-row function space into the count sector and the Johnson harmonic sectors. It also generated all the technical lemmas and initial proofs. The author's role was to pose the problem, guide the search direction, evaluate the AI-generated arguments, rewrite the proof, and take responsibility for the final form and validity of the result.